I am currently taking a Probability and Statistics course and learning the gamma function. Problems are usually phrased like "A group of scientists did x experiment and it followed a gamma distribution (...)" but I am wondering how do they know? Especially since the gamma function shape varies a lot depending on the values.

  • $\begingroup$ Gamma distribution emerges naturally in phenomena that follow the laws of the Poisson process. $\endgroup$
    – YJT
    Oct 30, 2020 at 14:01
  • $\begingroup$ @YJT so you mean that whenever we have an experiment with an event that happens on average x times every y interval then gamma distribution will occur naturally? $\endgroup$
    – Sergio
    Oct 30, 2020 at 14:05
  • $\begingroup$ The same way you can use a linear regression model and take a high correlation coefficient to suspect with reasonable certainty that the linear equation models the true data. $\endgroup$
    – user694818
    Oct 30, 2020 at 14:06
  • $\begingroup$ @MatthewDaly the case you are talking about is way simpler imho, do they first suspect it is a gamma distribution based on the shape and try to find values of alpha and beta by trial and error? $\endgroup$
    – Sergio
    Oct 30, 2020 at 14:10
  • $\begingroup$ That's not exactly the definition, but yes. For example, we know that radioactive elements emerge particles according to a Poisson process with some parameters. The time between turning on the equipment and detecting the first particle follows an exponential distribution. The time until the $n$th particle is already gamma (gamma=sum of exp). $\endgroup$
    – YJT
    Oct 30, 2020 at 14:12


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